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href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 变分推断</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%BF%91%E4%BC%BC%E8%B4%9D%E5%8F%B6%E6%96%AF%E8%AE%A1%E7%AE%97/"><i class="fa-fw fa-solid fa-cube"></i><span> 近似贝叶斯计算</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%A8%A1%E5%9E%8B%E6%AF%94%E8%BE%83%E4%B8%8E%E9%80%89%E6%8B%A9/"><i class="fa-fw fa-solid fa-ghost"></i><span> 模型比较与选择</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%B4%9D%E5%8F%B6%E6%96%AF%E4%BC%98%E5%8C%96/"><i class="fa-fw fa-solid fa-gas-pump"></i><span> 贝叶斯优化</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-ghost"></i><span> 不确定性DL</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/BayesNN/%E6%A6%82%E8%A7%88"><i class="fa-fw fa-solid fa-cube"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E5%8D%95%E4%B8%80%E7%A1%AE%E5%AE%9A%E6%80%A7%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 单一确定性神经网络</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C/"><i class="fa-fw fa-brands fa-deezer"></i><span> 贝叶斯神经网络</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E6%B7%B1%E5%BA%A6%E9%9B%86%E6%88%90/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 深度集成</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E6%95%B0%E6%8D%AE%E5%A2%9E%E5%BC%BA/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 数据增强</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E5%AF%B9%E6%AF%94%E4%B8%8E%E8%AF%84%E6%B5%8B/"><i class="fa-fw fa-brands fa-deezer"></i><span> 对比与评测</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-map"></i><span> 空间统计</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/GeoAI/%E7%BB%BC%E8%BF%B0%E7%B1%BB/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%82%B9%E5%8F%82%E8%80%83%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-solid fa-map"></i><span> 点参考数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E9%9D%A2%E5%85%83%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 面元数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%82%B9%E6%A8%A1%E5%BC%8F%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 点模式数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%96%B9%E6%B3%95/"><i class="fa-fw fa-solid fa-cube"></i><span> 空间贝叶斯方法</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E5%8F%98%E7%B3%BB%E6%95%B0%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-ghost"></i><span> 空间变系数模型</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E7%BB%9F%E8%AE%A1%E6%B7%B1%E5%BA%A6%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-brands fa-deezer"></i><span> 空间统计深度学习</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E6%97%B6%E7%A9%BA%E7%BB%9F%E8%AE%A1%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-atlas"></i><span> 时空统计模型</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E5%A4%A7%E6%95%B0%E6%8D%AE%E4%B8%93%E9%A2%98/"><i class="fa-fw fa fa-anchor"></i><span> 大数据专题</span></a></li><li><a class="site-page child" href="/categories/GeoAI/GeoAI/"><i class="fa-fw fa-brands fa-codepen"></i><span> GeoAI</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-database"></i><span> 基础</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 高等数学</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E6%A6%82%E7%8E%87%E4%B8%8E%E7%BB%9F%E8%AE%A1/"><i class="fa-fw fa-brands fa-deezer"></i><span> 概率与统计</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E7%BA%BF%E4%BB%A3%E4%B8%8E%E7%9F%A9%E9%98%B5%E8%AE%BA/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 线代与矩阵论</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E6%9C%80%E4%BC%98%E5%8C%96%E7%90%86%E8%AE%BA/"><i class="fa-fw fa-brands fa-codepen"></i><span> 最优化理论</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E4%BF%A1%E6%81%AF%E8%AE%BA/"><i class="fa-fw fa-solid fa-cube"></i><span> 信息论</span></a></li><li><a class="site-page child" href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0%E6%A8%A1%E5%9E%8B/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-ghost"></i><span> 机器学习</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E7%9F%A5%E8%AF%86%E5%9B%BE%E8%B0%B1/"><i class="fa-fw fa-solid fa-globe"></i><span> 知识图谱</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E8%87%AA%E7%84%B6%E8%AF%AD%E8%A8%80%E5%A4%84%E7%90%86/"><i class="fa-fw fa-solid 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href="https://xishansnow.github.io/ElementsOfStatisticalLearning/index.html"><i class="fa-fw fa-solid  fa-book-atlas"></i><span> 《统计学习精要（ESL）》</span></a></li><li><a class="site-page child" href="https://xishansnow.github.io/spatialSTAT_CN/index.html"><i class="fa-fw fa-solid  fa-layer-group"></i><span> 《空间统计学》</span></a></li><li><a class="site-page child" target="_blank" rel="noopener" href="https://otexts.com/fppcn/index.html"><i class="fa-fw fa-solid  fa-cloud-sun-rain"></i><span> 《预测：方法与实践》</span></a></li><li><a class="site-page child" href="https://xishansnow.github.io/MLAPP/index.html"><i class="fa-fw fa-solid  fa-robot"></i><span> 《机器学习的概率视角（MLAPP）》</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-compass"></i><span> 索引</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/archives/"><i class="fa-fw fa-solid fa-timeline"></i><span> 时间索引</span></a></li><li><a class="site-page child" href="/tags/"><i class="fa-fw fas fa-tags"></i><span> 标签索引</span></a></li><li><a class="site-page child" href="/categories/"><i class="fa-fw fas fa-folder-open"></i><span> 分类索引</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-link"></i><span> 其他</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/link/food/"><i class="fa-fw fas fa-utensils"></i><span> 美食博主</span></a></li><li><a class="site-page child" href="/link/photography"><i class="fa-fw fas fa-camera"></i><span> 摄影大神</span></a></li><li><a class="site-page child" href="/link/paper/"><i class="fa-fw fas fa-book-open"></i><span> 学术工具</span></a></li><li><a class="site-page child" href="/gallery/"><i class="fa-fw fas fa-images"></i><span> 摄影作品</span></a></li><li><a class="site-page child" href="/about/"><i class="fa-fw fas fa-heart"></i><span> 关于</span></a></li></ul></div></div></div></div><div class="post" id="body-wrap"><header class="post-bg" id="page-header" style="background-image: url('/img/007.png')"><nav id="nav"><span id="blog_name"><a id="site-name" href="/">西山晴雪的知识笔记</a></span><div id="menus"><div id="search-button"><a class="site-page social-icon search"><i class="fas fa-search fa-fw"></i><span> 搜索</span></a></div><div class="menus_items"><div class="menus_item"><a class="site-page" href="/"><i class="fa-fw fas fa-home"></i><span> 主页</span></a></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-atom"></i><span> 预测</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E5%B9%BF%E4%B9%89%E7%BA%BF%E6%80%A7%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-atom"></i><span> 广义线性模型</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E9%9D%9E%E5%8F%82%E6%95%B0%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-cogs"></i><span> 传统非参数模型</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E9%AB%98%E6%96%AF%E8%BF%87%E7%A8%8B/"><i class="fa-fw fas fa-school"></i><span> 高斯过程</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C/"><i class="fa-fw fas fa-layer-group"></i><span> 神经网络</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E6%A8%A1%E5%9E%8B%E9%80%89%E6%8B%A9%E4%B8%8E%E5%B9%B3%E5%9D%87/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 模型选择与平均</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E5%B0%8F%E6%A0%B7%E6%9C%AC%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-solid fa-globe"></i><span> 小样本学习</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-file-export"></i><span> 生成</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E4%BC%A0%E7%BB%9F%E6%A6%82%E7%8E%87%E5%9B%BE%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 传统概率图模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E6%9C%BA/"><i class="fa-fw fa-solid fa-deezer"></i><span> 玻耳兹曼机</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E5%8F%98%E5%88%86%E8%87%AA%E7%BC%96%E7%A0%81%E5%99%A8/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 变分自编码器</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E8%87%AA%E5%9B%9E%E5%BD%92%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-codepen"></i><span> 自回归模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E5%BD%92%E4%B8%80%E5%8C%96%E6%B5%81/"><i class="fa-fw fa-solid fa-cube"></i><span> 归一化流</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E6%89%A9%E6%95%A3%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-ghost"></i><span> 扩散模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E8%83%BD%E9%87%8F%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-gas-pump"></i><span> 能量模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E7%94%9F%E6%88%90%E5%BC%8F%E5%AF%B9%E6%8A%97%E7%BD%91%E7%BB%9C/"><i class="fa-fw fa-solid fa-globe"></i><span> 生成式对抗网络</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-magnet"></i><span> 挖掘</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E9%9A%90%E5%9B%A0%E5%AD%90%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 隐因子模型</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E7%8A%B6%E6%80%81%E7%A9%BA%E9%97%B4%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-deezer"></i><span> 状态空间模型</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E6%A6%82%E7%8E%87%E5%9B%BE%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 概率图学习</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E9%9D%9E%E5%8F%82%E6%95%B0%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-codepen"></i><span> 非参数贝叶斯模型</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E8%A1%A8%E7%A4%BA%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-solid fa-cube"></i><span> 表示学习</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E5%8F%AF%E8%A7%A3%E9%87%8A%E6%80%A7/"><i class="fa-fw fa-solid fa-ghost"></i><span> 可解释性</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E9%99%8D%E7%BB%B4/"><i class="fa-fw 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<link rel="stylesheet" type="text&#x2F;css" href="https://cdn.jsdelivr.net/hint.css/2.4.1/hint.min.css"><h1>机器学习和深度学习中常用的概率统计知识</h1>
<p>In Bayesian influence, probability distributions are heavily used to make intractable problems solvable. After discussing the <a target="_blank" rel="noopener" href="https://jonathan-hui.medium.com/normal-distributions-in-machine-learning-c8a21c8ba8c9">normal distribution</a>, we will cover other basic distributions and more advanced ones including Beta distribution, Dirichlet distribution, Poisson Distribution, and Gamma distribution. We will also discuss topics including the Conjugate prior, Exponential family of distribution, and Method of Moments.</p>
<h2 id="1-离散型分布">1 离散型分布</h2>
<h3 id="伯努利分布">伯努利分布</h3>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224553-dcf9.webp" alt=""></p>
<p>The Bernoulli distribution is a discrete distribution for a single binary random variable <em>X</em> ∈ {0, 1} with probability 1-<em>θ</em> and <em>θ</em> respectively. For example, when flipping a coin, the chance of head is <em>θ</em>.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224602-31a0.webp" alt=""></p>
<p>The expected value and the variance for the Bernoulli distribution are:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224608-4fad.webp" alt=""></p>
<h3 id="二项分布">二项分布</h3>
<p>The binomial distribution is the aggregated result of independent Bernoulli trials. For example, we flip a coin <em>N</em> times and model the chance to have <em>x</em> heads.</p>
<p><img src="https://miro.medium.com/max/1400/1*W0frbduJID1xCtGsfQ4HeA.png" alt=""></p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224613-ee07.webp" alt=""></p>
<p><a target="_blank" rel="noopener" href="https://en.wikipedia.org/wiki/Binomial_distribution">Source</a></p>
<p>The expected value and the variance for the binomial distribution are:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224617-adc2.webp" alt=""></p>
<h3 id="类别分布">类别分布</h3>
<p>A Bernoulli distribution has two possible outcomes. In a categorical distribution, we have <em>K</em> possible outcomes with probability <em>p₁, p₂, p₃, …</em> and <em>pk</em> accordingly. All these probabilities add up to one.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224621-ac8f.webp" alt=""></p>
<h3 id="多项分布">多项分布</h3>
<p>The multinomial distribution is a generalization of the binomial distributions. Instead of two outcomes, it has <em>k</em> possible outcomes. If binomial distribution is corresponding to the Bernoulli distribution, multinomial distribution is corresponding to categorical distribution.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224627-8490.webp" alt=""></p>
<p>Suppose these outcomes are associated with probabilities <em>θ₁</em>, <em>θ₂</em>, … and <em>θk</em> respectively. We collect a sample of size <em>N</em> and <em>xᵢ</em> represents the count for the outcome <em>i</em>. The joint probability is</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224632-5e15.webp" alt=""></p>
<p>The expected value and the variance for the multinomial distribution are:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224648-5d7e.webp" alt=""></p>
<h2 id="2-连续型分布">2 连续型分布</h2>
<h3 id="贝塔分布">贝塔分布</h3>
<p>For a Bernoulli distribution or a binomial distribution, how can we model the value for <em>θ</em>? For example, if a new virus is discovered, can we use a probability distribution to model the infection probability <em>θ</em>?</p>
<p>The beta distribution is a distribution over a continuous random variable on a finite interval of values. It is often used to model the probability for some binary event like <em>θ</em>. The model has two positive parameters <em>α</em> and <em>β</em> that affect the shape of the distribution.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224721-7b74.webp" alt=""></p>
<p><a target="_blank" rel="noopener" href="https://en.wikipedia.org/wiki/Beta_distribution">Source</a></p>
<p>When we have no knowledge about the new virus, we can set <em>α = β</em> = 1 for a uniform distribution, i.e. any possible probability values for <em>θ</em> ∈ [0, 1] are equally likely. This is our prior.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224707-0a1a.webp" alt=""></p>
<p><em>α = β</em> = 1 for uniform distribution</p>
<p>Then we can apply Bayes inference with the likelihood modeled by a binomial distribution. The posterior will be a beta distribution also with updates on <em>α</em> and <em>β</em>. This becomes the new infection rate distribution given the observed data and acts as the new prior when a new sample is observed.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224713-978c.webp" alt=""></p>
<p><a target="_blank" rel="noopener" href="https://en.wikipedia.org/wiki/Beta_distribution">Source</a></p>
<p>Mathematical, the beta distribution is defined as:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224716-ee4f.webp" alt=""></p>
<p>The beta function <em>B</em> normalized the R.H.S. to a probability distribution.</p>
<p>The definition seems complicated but when it is used in Bayesian inference, the calculation becomes very simple. Let’s say CDC reports <em>x</em> new infections out of <em>N</em> people. Applying the Bayes’ Theorem, the posterior will be:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224720-e8d2.webp" alt=""></p>
<p>i.e. we simply add the new positives to <em>α</em> and the new negatives (<em>N-x</em>) to <em>β</em>.</p>
<p>The expected value and variance for the beta distribution are</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224725-9b73.webp" alt=""></p>
<h3 id="狄利克雷分布">狄利克雷分布</h3>
<p>In the previous Bayesian inference example, the likelihood is modeled by the binomial distribution. We partner it with the beta distribution (prior) to calculate the posterior easily. For a likelihood with the multinomial distribution, the corresponding distribution is the Dirichlet distribution.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224728-eed0.webp" alt=""></p>
<p>Dirichlet distribution is defined as:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224732-3985.webp" alt=""></p>
<p>This random process has <em>K</em> outcomes and the corresponding Dirichlet distribution will be parameterized by a <em>K</em>-component <em>α</em>.</p>
<p>Similar to the beta distribution, its similarity with the corresponding likelihood makes the posterior computation easy.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224810-ea39.webp" alt=""></p>
<p>The expected value and the variance for the Dirichlet distribution are:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224805-716b.webp" alt=""></p>
<h3 id="泊松分布">泊松分布</h3>
<p>Poisson distribution models the probability for a given number of events occurring in a fixed interval of time. It models a Poisson process in which events occur independently and continuously at a constant average rate.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224817-6b03.webp" alt=""></p>
<p>As shown, a binomial distribution can be simplified to the Poisson distribution if the event is relatively rare.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224821-42a5.webp" alt=""></p>
<p>A Poisson process is assumed to be memoryless — the past does not influence any future predictions. The average wait time for the next event is the same regardless of whether the last event happened 1 minute or 5 hours ago.</p>
<p>The expected value and the variance for the Poisson distribution are:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224827-372b.webp" alt=""></p>
<h3 id="指数分布">指数分布</h3>
<p>The exponential distribution is the probability distribution for the waiting time before the next event occurs in a Poisson process. As shown in the right diagram below, for λ = 0.1 (<strong>rate parameter</strong>), the chance of waiting for more than 15 is 0.22.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224832-cea7.webp" alt=""></p>
<p>Mathematically, it is defined as:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224838-021c.webp" alt=""></p>
<p>The expected value and the variance for the exponential distribution are:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224843-15ca.webp" alt=""></p>
<h3 id="狄拉克分布">狄拉克分布</h3>
<p>The Dirac delta distribution (<em>δ</em> distribution) can be considered as a function that has a narrow peak at <em>x</em> = 0. Specifically, <em>δ</em>(<em>x</em>) has the value zero everywhere except at <em>x</em> = 0, and the area (integral) under the peak is 1.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224846-75a7.webp" alt=""></p>
<p>This function is a helpful approximation for a tall narrow spike function (an impulse) or some deterministic value in a probability distribution. It helps us to transform some models into mathematical equations.</p>
<h3 id="伽马分布">伽马分布</h3>
<p>The exponential distribution and the chi-squared distribution are special cases for the gamma distribution. The gamma distribution can be considered as the sum of <em>k</em> independent random variables with exponential distribution.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224907-2639.webp" alt=""></p>
<p>Intuitively, it is the distribution of the wait time for the _k_th events to occur.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224911-0854.webp" alt=""></p>
<p><a target="_blank" rel="noopener" href="https://en.wikipedia.org/wiki/Gamma_distribution">Source</a></p>
<p>Here is the mathematical definition for the gamma distribution.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224916-59ad.webp" alt=""></p>
<p>Depending on the context, the gamma distribution can be parameterized in two different ways.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224920-9463.webp" alt=""></p>
<p><em>α</em> (a.k.a. <em>k</em>) parameterizes the shape of the gamma distribution and <em>β</em> parameterizes the scale. As suggested by the Central Theorem, as <em>k</em> increases, the gamma distribution resembles the normal distribution.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224923-c3ed.webp" alt=""></p>
<p>As we change <em>β</em>, the shape remains the same but the scale of the <em>x</em> and <em>y</em>-axis change.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224925-12c0.webp" alt=""></p>
<p>The expectation and the variance of the Gamma distribution are:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224931-c7aa.webp" alt=""></p>
<h2 id="3-汇总">3 汇总</h2>
<p>Here is a recap of some of the probability distributions discussed.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224851-8900.webp" alt=""></p>
<p><a target="_blank" rel="noopener" href="http://www.cs.columbia.edu/~jebara/4771/tutorials/lecture12.pdf">Source</a></p>
<h2 id="4-共轭先验">4 共轭先验</h2>
<p>As discussed before, if we pair the distribution for the likelihood and the prior smartly, we can turn the Bayesian inference to be tractable.</p>
<p>In Bayesian inference, a prior is a conjugate prior if the corresponding posterior belongs to the same class of distribution of the prior.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224937-8655.webp" alt=""></p>
<p>For example, the beta distribution is a conjugate prior to the binomial distribution (likelihood). The calculated posterior with the Bayes’ Theorem is a beta distribution also. Here are more examples of conjugate priors.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224940-6f5b.webp" alt=""></p>
<p><a target="_blank" rel="noopener" href="https://en.wikipedia.org/wiki/Conjugate_prior">Source</a></p>
<h2 id="5-充分统计量">5 充分统计量</h2>
<p>By definition, when a distribution is written in the form of</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224949-5a22.webp" alt=""></p>
<p><em>T</em>(<em>x</em>) is called sufficient statistics.</p>
<p>Here is an example applied to the Poisson distribution.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224946-6259.webp" alt=""></p>
<p>T(<em>x</em>) sums over _x_ⱼ.</p>
<p>The significance of sufficient statistics is that no other statistic calculated from <em>x₁, x₂, x₃, …</em> will provide any additional information to estimate the distribution parameter <em>θ</em>. If we know <em>T</em>(<em>x</em>), we have sufficient information to estimate <em>θ</em>. No other information is needed. We don’t need to keep <em>x₁, x₂, x₃, …</em> around to build the model. For example, given a Poisson distribution modeled by <em>θ</em> (a.k.a. λ), we can estimate <em>θ</em> by dividing <em>T</em>(<em>x</em>) with <em>n</em>.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224952-6828.webp" alt=""></p>
<h2 id="6-指数族分布">6 指数族分布</h2>
<p>Normal, Bernoulli, gamma, beta, Dirichlet, exponential, Poisson distribution, and many other distributions belong to a family of distribution called the exponential family. It has the form of</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224955-7ee9.webp" alt=""></p>
<p>Here are the exponential family forms, represented by <em>h</em>(<em>x</em>), <em>η, T</em>(<em>x</em>), and <em>A,</em> for the binomial and Poisson distribution.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320224957-2dcc.webp" alt=""></p>
<p>Modified from <a target="_blank" rel="noopener" href="https://ocw.mit.edu/courses/mathematics/18-655-mathematical-statistics-spring-2016/lecture-notes/MIT18_655S16_LecNote7.pdf">source</a></p>
<p>We can convert parameter <em>θ</em> and the natural parameter <em>η</em> from each other. For example, the Bernoulli parameter <em>θ</em> can be calculated from the corresponding natural parameter <em>η</em> using the logistic function.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225000-a348.webp" alt=""></p>
<p>Here is another example in writing the normal distribution in the form of an exponential family.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225003-41c5.webp" alt=""></p>
<p>What is the advantage of this abstract generalization?</p>
<p>The exponential family provides a general mathematical framework in solving problems for its family of distributions. For example, computing the expected value for the Poisson distribution can be hard.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225005-d714.webp" alt=""></p>
<p>Instead, all the expected values for the exponential family can be calculated fairly easily for <em>A</em>. As shown on the left below, A’(<em>η</em>) equals the expected value for <em>T</em>(<em>x</em>). Since <em>T</em>(<em>x</em>) = <em>x</em> and <em>λ</em> = exp(<em>η</em>) and <em>A</em>(<em>λ) = λ =</em> exp(<em>η</em>) in the Poisson distribution, we differentiate A(<em>η</em>) to find 𝔼[<em>x</em>]. This equals <em>λ</em>.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225008-9af7.webp" alt=""></p>
<p>This family of distribution has nice properties in Bayesian analysis also. If the likelihood belongs to an exponential family, there exists a conjugate prior that is often an exponential family. If we have an exponential family written as</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225011-14e5.webp" alt=""></p>
<p>the conjugate prior parameterized by γ will have the form</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225013-c76a.webp" alt=""></p>
<p>The conjugate prior, modeled by <em>γ</em>, will have one additional degree of freedom. For example, the Bernoulli distribution has one degree of freedom modeled by <em>θ</em>. The corresponding beta distribution will have two degrees of freedom modeled by <em>α</em> and <em>β</em>.</p>
<p>Consider the Bernoulli distribution below in the form of the exponential family,</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225015-3687.webp" alt=""></p>
<p>We can define (or guess)</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225017-3090.webp" alt=""></p>
<p>We get</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225018-376b.webp" alt=""></p>
<p>i.e. beta distribution is a conjugate prior to the Bernoulli distribution.</p>
<h2 id="7-最大熵原则">7 最大熵原则</h2>
<p>There are possibly infinite models that can fit the prior data (prior knowledge) exactly. The principle of maximum entropy asserts that the probability distribution that best represents a system is the one with the largest entropy. In information theory, the entropy of a random variable measures the “surprise” inherent to the possible outcomes. Under this principle, we avoid applying unnecessary and additional constraints on what is possible, as constraints decrease the entropy of the system.</p>
<p>Many distributions can satisfy the constraints imposed by sufficient statistics. But the one that we may choose is the one with the highest entropy. It can be proven that the exponential family has the maximum-entropy distribution consistent with the given constraints on sufficient statistics.</p>
<h2 id="8-K-阶矩">8 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span> 阶矩</h2>
<p>A moment describes the shape of a function quantitatively. If the function <em>f</em> is a probability distribution, the zero moment is the total probability (=1), the first moment is the mean. For the 2nd and higher moments, the central moments provide better information about the distribution’s shape. The second central moment is the variance, the third standardized moment is the skewness, and the fourth moment is the kurtosis.</p>
<p>为了定量地描述概率分布的形状，科学家提出了 <strong>矩（ Moment ）</strong> 的概念。矩是一个标量，不同阶的矩大小通常反映了概率分布形状的某一种特性。</p>
<p>如果函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 是随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">X</mi></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">X</span></span></span></span> 的概率分布，则：</p>
<ul>
<li><code>零阶矩</code> 为总概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li>
<li><code>一阶矩</code>为均值 (Mean)，反映了分布的中心位置，定义为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>e</mi><mi>a</mi><mi>n</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mtext>）</mtext><mo>=</mo><mi>μ</mi><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>x</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Mean(\mathbf{X}）=\mu = \int_{-\infty}^{+\infty} x f(x)dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal">e</span><span class="mord mathnormal">an</span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mord cjk_fallback">）</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3804em;vertical-align:-0.4142em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9662em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4142em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span>。</li>
</ul>
<p>在一阶矩基础上，通常采用中心矩的方式来定义其他矩，以便更好地提供分布形状的信息。</p>
<ul>
<li><code>一阶中心距</code> 显然为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li>
<li><code>二阶中心矩</code> 为方差 (Variance)，反映了分布的离散度（或反之，聚集程度），定义为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mi mathvariant="double-struck">E</mi><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi mathvariant="double-struck">E</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="true">]</mo></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msup><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><mi mathvariant="double-struck">E</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow><mn>2</mn></msup><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Var(\mathbf{X})=\sigma^2= \mathbb{E}\left[(x-\mathbb{E}(\mathbf{X}))^2\right] = \int_{-\infty}^{+\infty}\left( x-\mathbb{E}(\mathbf{X} ) \right)^2 \,f(x) \,dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Va</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathbb">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbb">E</span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3804em;vertical-align:-0.4142em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9662em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4142em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbb">E</span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></li>
</ul>
<p>在二阶中心矩基础上，可以对更高阶的矩做归一化处理，得到高阶的标准化矩：</p>
<ul>
<li><code>三阶标准化距</code>为偏度 (Skewness)，反映了分布的对称程度，定义为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>k</mi><mi>e</mi><mi>w</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="double-struck">E</mi><mrow><mo fence="true">[</mo><msup><mrow><mo fence="true">(</mo><mfrac><mrow><mi mathvariant="bold">X</mi><mo>−</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac><mo fence="true">)</mo></mrow><mn>3</mn></msup><mo fence="true">]</mo></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msup><mrow><mo fence="true">(</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi mathvariant="double-struck">E</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo></mrow><mi>σ</mi></mfrac><mo fence="true">)</mo></mrow><mn>3</mn></msup><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Skew(\mathbf{X}) = \mathbb{E}\left[ \left(\frac{\mathbf{X}-\mu}{\sigma} \right)^3 \right] = \int_{-\infty}^{+\infty}\left(\frac{x-\mathbb{E}(\mathbf{X})}{\sigma} \right)^3 \,f(x) \,dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord mathbb">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9264em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">X</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.354em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.004em;vertical-align:-0.65em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9662em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4142em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathbb mtight">E</span><span class="mopen mtight">(</span><span class="mord mathbf mtight">X</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.354em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span> 。偏度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 时，为对称分布；偏度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 时为右偏分布；偏度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 时为左偏分布。</li>
<li><code>四阶标准化距</code>为峰度 (Kurtosis)，反映了分布中峰值的尖度，定义为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>u</mi><mi>r</mi><mi>t</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="double-struck">E</mi><mrow><mo fence="true">[</mo><msup><mrow><mo fence="true">(</mo><mfrac><mrow><mi mathvariant="bold">X</mi><mo>−</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac><mo fence="true">)</mo></mrow><mn>4</mn></msup><mo fence="true">]</mo></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msup><mrow><mo fence="true">(</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi mathvariant="double-struck">E</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo></mrow><mi>σ</mi></mfrac><mo fence="true">)</mo></mrow><mn>4</mn></msup><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Kurt(\mathbf{X})= \mathbb{E}\left[ \left(\frac{\mathbf{X}-\mu}{\sigma} \right)^4 \right] = \int_{-\infty}^{+\infty}\left( \frac{x-\mathbb{E}(\mathbf{X} )}{\sigma} \right)^4 \,f(x) \,dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal">u</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord mathbb">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9264em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">X</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.354em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.004em;vertical-align:-0.65em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9662em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4142em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathbb mtight">E</span><span class="mopen mtight">(</span><span class="mord mathbf mtight">X</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.354em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></li>
</ul>
<p>在单变量矩以及中心矩基础上，可以定义两个随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">X</mi></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Y</mi></mrow><annotation encoding="application/x-tex">\mathbf{Y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.02875em;">Y</span></span></span></span>之间的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>+</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">k+p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 阶混合中心距 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mi mathvariant="bold">X</mi><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo><mi>k</mi></msup><mo stretchy="false">(</mo><mi mathvariant="bold">Y</mi><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mi mathvariant="bold">Y</mi><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo><mi>p</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">{E[ (\mathbf{X}-E[\mathbf{X}] )^k( \mathbf{Y}-E[\mathbf{Y}] )^p]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">[(</span><span class="mord mathbf">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">[</span><span class="mord mathbf">X</span><span class="mclose">]</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">[</span><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="mclose">]</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span> 。</p>
<p>常用的混合中心矩为（1+1）阶混合中心距，即协方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">Y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mi mathvariant="bold">X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold">Y</mi><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mi mathvariant="bold">Y</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex">cov(\mathbf{X},\mathbf{Y})={E[ (\mathbf{X}-E[\mathbf{X}] )( \mathbf{Y}-E[\mathbf{Y}] )]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">co</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">[(</span><span class="mord mathbf">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">[</span><span class="mord mathbf">X</span><span class="mclose">])</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">[</span><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="mclose">])]</span></span></span></span></span></p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225022-3a4a.webp" alt=""></p>
<p><a target="_blank" rel="noopener" href="https://www.researchgate.net/figure/Illustration-of-the-skewness-and-kurtosis-values-and-how-they-correlate-with-the-shape-of_fig1_298415862">Source</a></p>
<p>The _k_th moment, or the _k_th raw moment, of function <em>f</em> is defined as</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225024-9b79.webp" alt=""></p>
<p>This moment is called the moment about zero. But if we subtract <em>x</em> with the mean first, it will be called a central moment.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225039-fbd4.webp" alt=""></p>
<p>The _k_th moment equals the _k_th-order derivative of A(η).</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225030-8748.webp" alt=""></p>
<h2 id="9-矩的使用">9 矩的使用</h2>
<p>现在考虑一种场景，根据矩的定义，当概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 比较简单时（如常见的高斯等分布），有些能够得到矩的解析解。但现实大部分场景中，随机变量的分布是复杂的，通常无法得到解析解，只能通过样本来获得近似的数值解。那么如何通过样本来估计概率分布（模型）的参数呢？或者说，如何用样本分布 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>q</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">q^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 对总体分布 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 建模呢？</p>
<p>有了对矩的认识，我们就可以从样本数据中计算矩，以使两者的充分统计量的期望能够匹配。</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225046-834d.webp" alt=""></p>
<p>Consider a simple zero-centered distribution model <em>f</em> parameterized by <em>θ</em> with <em>T</em>(<em>X</em>)<em>=x</em>.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225049-c05c.webp" alt=""></p>
<p>The first and second theoretical moment is:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225053-fe75.webp" alt=""></p>
<p>Modified from <a target="_blank" rel="noopener" href="http://people.missouristate.edu/songfengzheng/Teaching/MTH541/Lecture%20notes/MOM.pdf">source</a></p>
<p>The second-order sample moment is:</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225055-6bf1.webp" alt=""></p>
<p>By letting the sample moment equal to the theoretical moment, we get an estimation of <em>σ</em> (sampled <em>σ</em>) as.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225059-b80c.webp" alt=""></p>
<p>But the integration is not easy in general. But we can use the derivatives of <em>A</em> to compute the moment and solve the distribution parameter. For example, in the gamma distribution, its parameters <em>α</em> and <em>β</em> can be estimated from the sample mean and variance.</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20220320225101-cbd9.webp" alt=""></p>
</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta">文章作者: </span><span class="post-copyright-info"><a href="http://xishansnow.github.io">西山晴雪</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta">文章链接: </span><span class="post-copyright-info"><a href="http://xishansnow.github.io/posts/15a1fc08.html">http://xishansnow.github.io/posts/15a1fc08.html</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta">版权声明: </span><span class="post-copyright-info">本博客所有文章除特别声明外，均采用 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank">CC BY-NC-SA 4.0</a> 许可协议。转载请注明来自 <a href="http://xishansnow.github.io" target="_blank">西山晴雪的知识笔记</a>！</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/tags/GeoAI/">GeoAI</a><a class="post-meta__tags" 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class="title">从自然语言文本中收割地理空间大数据</div></div></a></div></div></div></div><div class="aside-content" id="aside-content"><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>目录</span><span class="toc-percentage"></span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-text">机器学习和深度学习中常用的概率统计知识</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#1-%E7%A6%BB%E6%95%A3%E5%9E%8B%E5%88%86%E5%B8%83"><span class="toc-text">1 离散型分布</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%BC%AF%E5%8A%AA%E5%88%A9%E5%88%86%E5%B8%83"><span class="toc-text">伯努利分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%BA%8C%E9%A1%B9%E5%88%86%E5%B8%83"><span class="toc-text">二项分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%B1%BB%E5%88%AB%E5%88%86%E5%B8%83"><span class="toc-text">类别分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%A4%9A%E9%A1%B9%E5%88%86%E5%B8%83"><span class="toc-text">多项分布</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#2-%E8%BF%9E%E7%BB%AD%E5%9E%8B%E5%88%86%E5%B8%83"><span class="toc-text">2 连续型分布</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E8%B4%9D%E5%A1%94%E5%88%86%E5%B8%83"><span class="toc-text">贝塔分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%8B%84%E5%88%A9%E5%85%8B%E9%9B%B7%E5%88%86%E5%B8%83"><span class="toc-text">狄利克雷分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%B3%8A%E6%9D%BE%E5%88%86%E5%B8%83"><span class="toc-text">泊松分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%8C%87%E6%95%B0%E5%88%86%E5%B8%83"><span class="toc-text">指数分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%8B%84%E6%8B%89%E5%85%8B%E5%88%86%E5%B8%83"><span class="toc-text">狄拉克分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%BC%BD%E9%A9%AC%E5%88%86%E5%B8%83"><span class="toc-text">伽马分布</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#3-%E6%B1%87%E6%80%BB"><span class="toc-text">3 汇总</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#4-%E5%85%B1%E8%BD%AD%E5%85%88%E9%AA%8C"><span class="toc-text">4 共轭先验</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#5-%E5%85%85%E5%88%86%E7%BB%9F%E8%AE%A1%E9%87%8F"><span class="toc-text">5 充分统计量</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#6-%E6%8C%87%E6%95%B0%E6%97%8F%E5%88%86%E5%B8%83"><span class="toc-text">6 指数族分布</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#7-%E6%9C%80%E5%A4%A7%E7%86%B5%E5%8E%9F%E5%88%99"><span class="toc-text">7 最大熵原则</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#8-K-%E9%98%B6%E7%9F%A9"><span class="toc-text">8 KKK 阶矩</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#9-%E7%9F%A9%E7%9A%84%E4%BD%BF%E7%94%A8"><span class="toc-text">9 矩的使用</span></a></li></ol></li></ol></div></div></div></div></main><footer id="footer"><div id="footer-wrap"><div class="copyright">&copy;2020 - 2023 By 西山晴雪</div><div class="framework-info"><span>框架 </span><a target="_blank" rel="noopener" href="https://hexo.io">Hexo</a><span class="footer-separator">|</span><span>主题 </span><a target="_blank" rel="noopener" href="https://github.com/jerryc127/hexo-theme-butterfly">Butterfly</a></div></div></footer></div><div id="rightside"><div id="rightside-config-hide"><button id="readmode" type="button" title="阅读模式"><i class="fas fa-book-open"></i></button><button id="translateLink" type="button" title="简繁转换">繁</button><button id="darkmode" type="button" 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